Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by B, F/Left Cosets

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Examples of Left Cosets of Subgroups of Dihedral Group $D_4$

Let the dihedral group $D_4$ be represented by the set of square matrices:

$D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$


under the operation of conventional matrix multiplication, where:

$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$

$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}

\qquad \mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \qquad \mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$


Let $H \subseteq D_4$ be defined as:

$H = \gen {\mathbf B, \mathbf F}$

where $\gen {\mathbf B, \mathbf F}$ denotes the subgroup generated by $\set {\mathbf B, \mathbf F}$.

From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf B, \mathbf F}$ we have:

$\gen {\mathbf B, \mathbf F} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$


The left cosets of $H$ are:

\(\ds \mathbf I H\) \(=\) \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \mathbf B H\)
\(\ds \) \(=\) \(\ds \mathbf D H\)
\(\ds \) \(=\) \(\ds \mathbf F H\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf A H\) \(=\) \(\ds \set {\mathbf A, \mathbf C, \mathbf E, \mathbf G}\)
\(\ds \) \(=\) \(\ds \mathbf C H\)
\(\ds \) \(=\) \(\ds \mathbf E H\)
\(\ds \) \(=\) \(\ds \mathbf G H\)


Proof

The Cayley table of $D_4$ is presented as:

$\begin{array}{r|rrrrrrrr}
         & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\

\hline \mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \mathbf A & \mathbf A & \mathbf B & \mathbf C & \mathbf I & \mathbf E & \mathbf F & \mathbf G & \mathbf D \\ \mathbf B & \mathbf B & \mathbf C & \mathbf I & \mathbf A & \mathbf F & \mathbf G & \mathbf D & \mathbf E \\ \mathbf C & \mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf G & \mathbf D & \mathbf E & \mathbf F \\ \mathbf D & \mathbf D & \mathbf G & \mathbf F & \mathbf E & \mathbf I & \mathbf C & \mathbf B & \mathbf A \\ \mathbf E & \mathbf E & \mathbf D & \mathbf G & \mathbf F & \mathbf A & \mathbf I & \mathbf C & \mathbf B \\ \mathbf F & \mathbf F & \mathbf E & \mathbf D & \mathbf G & \mathbf B & \mathbf A & \mathbf I & \mathbf C \\ \mathbf G & \mathbf G & \mathbf F & \mathbf E & \mathbf D & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$


Thus:

\(\ds \mathbf I H\) \(=\) \(\ds \mathbf I \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf I^2, \mathbf I \mathbf B, \mathbf I \mathbf D, \mathbf I \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf B H\) \(=\) \(\ds \mathbf B \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf B \mathbf I, \mathbf B^2, \mathbf B \mathbf D, \mathbf B \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf B, \mathbf I, \mathbf F, \mathbf D}\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf D H\) \(=\) \(\ds \mathbf D \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf D \mathbf I, \mathbf D \mathbf B, \mathbf D^2, \mathbf D \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf D, \mathbf F, \mathbf I, \mathbf B}\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf F H\) \(=\) \(\ds \mathbf F \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf F \mathbf I, \mathbf F \mathbf B, \mathbf F \mathbf D, \mathbf F^2}\)
\(\ds \) \(=\) \(\ds \set {\mathbf F, \mathbf D, \mathbf B, \mathbf I}\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf A H\) \(=\) \(\ds \mathbf A \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf A \mathbf I, \mathbf A \mathbf B, \mathbf A \mathbf D, \mathbf A \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf A, \mathbf C, \mathbf E, \mathbf G}\)


\(\ds \mathbf C H\) \(=\) \(\ds \mathbf C \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf C \mathbf I, \mathbf C \mathbf B, \mathbf C \mathbf D, \mathbf C \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf C, \mathbf A, \mathbf G, \mathbf E}\)
\(\ds \) \(=\) \(\ds \mathbf A H\)


\(\ds \mathbf E H\) \(=\) \(\ds \mathbf E \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf E \mathbf I, \mathbf E \mathbf B, \mathbf E \mathbf D, \mathbf E \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf E, \mathbf G, \mathbf A, \mathbf C}\)
\(\ds \) \(=\) \(\ds \mathbf A H\)


\(\ds \mathbf G H\) \(=\) \(\ds \mathbf G \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf G \mathbf I, \mathbf G \mathbf B, \mathbf G \mathbf D, \mathbf G \mathbf F}\)
\(\ds \) \(=\) \(\ds \set {\mathbf G, \mathbf E, \mathbf C, \mathbf A}\)
\(\ds \) \(=\) \(\ds \mathbf A H\)

$\blacksquare$


Sources

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